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The aim of the investigation is to find differences of small n x n squares in 10 x 10 square and then to see if there is any rule or pattern which connects the size of square chosen and the difference.

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Introduction

Pattens in Squares

The aim of the investigation is to find differences of small n x n squares in 10 x 10 square and then to see if there is any rule or pattern which connects the size of square chosen and the difference.

In order to find the difference of n x n square first step is to take  nxn square and then multiply the corners diagonally.

Middle

2 +11n - n2 +10n + n+10

11     12

n+10   n+11                                                        n2 +11n – n2 +11n +10

11 x 2 = 22                                                        10

'font-size:14.0pt; '>12 x 1 = 12

'font-size:14.0pt; '>22 – 12 = 10

In both ways the difference is 10

3x3 square

n         n+2                                                           n(n+22) – [(n+2) (n+20)]

1     2    3

11  12  13                                                            n2+22n - n2+20n +2n +40

21  22  23

n+20   n+22                                                         n2 +22n - n2 +22n +40

21 x 3 = 63                                                          40

23 x 1 = 23

63 – 23 = 40

In both ways the difference is 40

Conclusion

60;

31 32  33  34                                                      n2+33n - n2+33n + 90

n+30         n+33                                                  10

31 x 4 = 124                In both ways the difference is 90

34 x 1 = 31

124 – 34 = 90

5x5 square

n                 n+4                                                n(n+44) - [(n+4) (n+40)]

1    2   3    4   5

11 12 13 14 15                                                 n2+44n - n2+40n + 4n +160

21 22 23 24 25

31 32 33 34 35                                                 n2+44n - n2+44n +160

41 42 43 44 45

n+40           n+44                                              160

41 x 5 = 205

45 x 1 = 45

205 – 45 = 160

In both ways the difference is 160

6x6 square

n                       n+5                                          n(n+55) - [(n+5) (n+50)

1   2   3   4   5    6

11 12 13 14 15 16                                           n2+55n - n2+50n + 5n + 250

21 22 23 24 25 26

31 32 33 34 35 36                                           n2+55n - n2+55n + 250

41 42 43 44 45 46

51 52 53 54 55 56                                           250

n+50                 n+55

51 x 6 = 306

56 x 1 = 56

306 – 56 = 250

In both ways the difference is 250

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